Very ample linear series on quadrigonal curves

Coppens, Marc; Martens, Gerriet

doi:10.36045/j.bbms.210506a

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Reduction To Castelnuovo Curves1

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2023

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“…The genus bound in Theorem 4.1 is not the best possible; there may exist extremal series on C which we did not consider in the proof of the Theorem. For instance, for k = 4 we need no bound at all (as is also seen from [9]), for k = 5 we must exclude only g = 14 and for k = 6 only g = 20, g = 25 and g = 35. In fact, a refinement of the discussion in the proof of Theorem 4.1 shows that we may replace the genus bound g…”

Section: Reduction To Castelnuovo Curvesmentioning

confidence: 99%

On curves on Hirzebruch surfaces

Martens

2023

Abh. Math. Semin. Univ. Hambg.

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We call a smooth irreducible projective curve a Castelnuovo curve if it admits a birational map into the projective r-space such that the image curve has degree at least 2r+1 and the maximum possible geometric genus (which one can calculate by a classical formula due to Castelnuovo). It is well known that a Castelnuovo curve must lie on a Hirzebruch surface (rational ruled surface). Conversely, making use of a result of W. Castryck and F. Cools concerning the scrollar invariants of curves on Hirzebruch surfaces we show that curves on Hirzebruch surfaces are Castelnuovo curves unless their genus becomes too small w.r.t. their gonality. We analyze the situation more closely, and we calculate the number of moduli of curves of fixed genus g and fixed gonality k lying on Hirzebruch surfaces, in terms of g and k.

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Section: Reduction To Castelnuovo Curvesmentioning

confidence: 99%